sigma09-014. 301KB Jun 04 2011 12:10:23 AM

Symmetry, Integrability and Geometry: Methods and Applications

SIGMA 5 (2009), 014, 17 pages

Simple Finite Jordan Pseudoalgebras⋆
Pavel KOLESNIKOV
Sobolev Institute of Mathematics, 4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia
E-mail: pavelsk@math.nsc.ru
Received September 12, 2008, in final form January 10, 2009; Published online January 30, 2009
doi:10.3842/SIGMA.2009.014
Abstract. We consider the structure of Jordan H-pseudoalgebras which are linearly finitely
generated over a Hopf algebra H. There are two cases under consideration: H = U (h) and
H = U (h)#C[Γ], where h is a finite-dimensional Lie algebra over C, Γ is an arbitrary group
acting on U (h) by automorphisms. We construct an analogue of the Tits–Kantor–Koecher
construction for finite Jordan pseudoalgebras and describe all simple ones.
Key words: Jordan pseudoalgebra; conformal algebra; TKK-construction
2000 Mathematics Subject Classification: 17C50; 17B60; 16W30

1

Introduction

The notion of pseudoalgebra appeared as a natural generalization of the notion of conformal
algebra. The last one provides a formal language describing algebraic structures underlying
the singular part of the operator product expansion (OPE) in conformal field theory. Roughly
speaking, the OPE of two local chiral fields is a formal distribution in two variables presented
NP
−1
cn (z)(w − z)−n−1 [3]. The coefficients cn , n ∈ Z, of this distribution are considered
as
n=−∞

as new “products” on the space of fields. The algebraic system obtained is called a vertex
algebra. Its formal axiomatic description was stated in [4] (see also [7]). The “singular part”
of a vertex algebra, i.e., the structure defined by only those operations with non-negative n, is
a (Lie) conformal algebra [7].
Another approach to the theory of vertex algebras gives rise to the notion of a pseudotensor
category [2] (which is similar to the multicategory of [14]). Given a Hopf algebra H, one may
define the pseudotensor category M∗ (H) [1] (objects of this category are left H-modules). An
algebra in this category is called an H-pseudoalgebra. A pseudoalgebra is said to be finite if it
is a finitely generated H-module.

In particular, for the one-dimensional Hopf algebra H = k, k is a field, an H-pseudoalgebra
is just an ordinary algebra over the field k. For H = k[D], an H-pseudoalgebra is exactly the
same as conformal algebra. In a more general case H = k[D1 , . . . , Dn ], n ≥ 2, the notion of
an H-pseudoalgebra is closely related with Hamiltonian formalism in the theory of non-linear
evolution equations [1]. For an arbitrary Hopf algebra H, an H-pseudoalgebra defines a functor
from the category of H-bimodule associative commutative algebras to the category of H-module
algebras (also called H-differential algebras).
An arbitrary conformal algebra C can be canonically embedded in a “universal” way into
the space of formal power series A[[z, z −1 ]] over an appropriate ordinary algebra A [7, 15]. This
algebra A = Coeff C is called the coefficient algebra of C. A conformal algebra is said to be
associative (Lie, Jordan, etc.) if so is its coefficient algebra. For pseudoalgebras, a construction called annihilation algebra [1] works instead of coefficient algebra. However, the notion of
⋆

This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The full collection
is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html

2

P. Kolesnikov

a pseudotensor category provides a direct way to the definition of what is a variety of pseudoalgebras [13].
In the paper [5], the complete description of simple finite Lie conformal algebras over k = C
was obtained. Apart from current conformal algebras, the only example of a simple finite
Lie conformal algebra is the Virasoro conformal algebra. In the associative case, there are no
exceptional examples: A simple finite associative conformal algebra is isomorphic to the current
algebra over Mn (C), n ≥ 1. It was shown in [19] that there are no exceptional examples of
simple finite Jordan conformal algebras.
In [1], the structure theory of finite Lie pseudoalgebras was developed. The classification
theorem [1, Theorem 13.2] states that there exist simple finite Lie pseudoalgebras which are
not isomorphic to current pseudoalgebras over ordinary simple finite-dimensional Lie algebras.
This is not the case for associative pseudoalgebras. In this paper, we show the same for finite
simple Jordan pseudoalgebras (Theorem 3): There are no examples of such pseudoalgebras
except for current algebras (if H = U (h)) or transitive direct sums of current algebras (if
H = U (h)#C[Γ]). The main tool of the proof is an analogue of the well known Tits–Kantor–
Koecher (TKK) construction for Jordan algebras. This result generalizes the classification of
finite Jordan conformal algebras [19] to “multi-dimensional” case.
It was shown in [9] that the structure theory of Jordan conformal superalgebras is richer.
The classification of simple finite Jordan superalgebras based on the structure theory of finite
conformal Lie superalgebras [6] includes one series and two exceptional algebras [9, Theorem 3.9].
In our proof, we will not use annihilation algebras directly, the TKK construction will be

built on the level of pseudoalgebras.
The paper is organized as follows. Section 2 contains the basics of Hopf algebras and pseudoalgebras theory, and some notations that will be used later. In Section 3, we introduce an analogue
of the Tits–Kantor–Koecher construction (TKK) for finite Jordan pseudoalgebras. To complete
the classification of finite simple Jordan pseudoalgebras, we need some technical results proved
in Section 4. The main case under consideration is H = U (h), where h is a finite-dimensional Lie
algebra over C. Another case is the smash-product U (h)#C[Γ], where Γ is an arbitrary group.
These cases describe all cocommutative Hopf algebras over C with finite-dimensional spaces of
primitive elements (see, e.g., [17]).

2

Preliminaries on Hopf algebras and pseudoalgebras

2.1

Hopf algebras

In this section, we state some notations that will be used later.
An associative algebra H with a unit (over a field k) endowed with coassociative coproduct
∆ : H → H ⊗ H and counit ε : H → k is called a bialgebra. Recall that both ∆ and ε are

homomorphisms of algebras and
(id ⊗∆)∆(h) = (∆ ⊗ id)∆(h),

(ε ⊗ id)∆(h) = (id ⊗ε)∆(h) = h.

To simplify the notation,
we will use the following one which P
is due to Sweedler [17]: ∆[1] (h) := h,
P
[2]
[n]
[n−1]
∆ (h) := ∆(h) = h(1) ⊗ h(2) , ∆ (h) := (id ⊗∆
) = h(1) ⊗ · · · ⊗ h(n) . Further, we will
(h)
(h)
P
[n]
omit the symbol
by writing ∆(h) = h(1) ⊗ h(2) , ∆ (h) = h(1) ⊗ · · · ⊗ h(n) , etc.

(h)

Given a bialgebra H, an antihomomorphism S : H → H is called an antipode, if it satisfies
S(h(1) )h(2) = ε(h) = h(1) S(h(2) ).

A bialgebra with an antipode is called a Hopf algebra.

Simple Finite Jordan Pseudoalgebras

3

There exists a natural structure of (right) H-module on the nth tensor power of H (denoted
by H ⊗n ):
(f1 ⊗ · · · ⊗ fn ) · h = f1 h(1) ⊗ · · · ⊗ fn h(n) ,

fi , h ∈ H.

(2.1)

In this paper, we substantially consider cocommutative Hopf algebras, i.e., such that h(1) ⊗

h(2) = h(2) ⊗ h(1) for all h ∈ H. The antipode S of a cocommutative Hopf algebra is involutive,
i.e., S 2 = id.
For example, the universal enveloping algebra U (g) of a Lie algebra g over a field of zero
characteristic is a cocommutative Hopf algebra. Another series of examples is provided by the
group algebra k[Γ] of an arbitrary group Γ and by the general construction of smash-product.
Namely, suppose H is a Hopf algebra, and let a group Γ acts on H by algebra automorphisms.
Then one may define the following new product on H ⊗ k[Γ]:
(h1 ⊗ g1 ) · (h2 ⊗ g2 ) = h1 hg21 ⊗ g1 g2 .
The algebra obtained is denoted by H#k[Γ]. Together with usual coproduct and antipode
defined as on H ⊗ k[Γ], H#k[Γ] is a Hopf algebra (the smash product of H and k[G]). If H
is cocommutative, then so is H#k[Γ]. Moreover, if k is an algebraically closed field of zero
characteristic, then every cocommutative Hopf algebra H over k is isomorphic to the smash
product U (g)#k[Γ] for appropriate g and Γ [16].
Lemma 1 ([1]). Let H be a cocommutative Hopf algebra, and let {hi | i ∈ I} be a linear basis
of H. Then every element F ∈ H ⊗n+1 , n ≥ 1, can be uniquely presented as
X
(hi1 ⊗ · · · ⊗ hin ⊗ 1) · gi1 ,...,in ,
F =
gi1 ,...,in ∈ H.
(2.2)

i1 ,...,in

In other words, the set {hi1 ⊗ · · · ⊗ hin ⊗ 1 | i1 , . . . , in ∈ I} is an H-basis of the H-module
H ⊗n+1 (2.1).
To find the presentation (2.2), one may use formal Fourier transformation F and its inverse F −1 [1]:
F, F −1 : H ⊗n+1 → H ⊗n+1 ,
F : h1 ⊗ · · · ⊗ hn ⊗ f 7→ h1 f(1) ⊗ · · · ⊗ hn f(n) ⊗ f(n+1) ,
F −1 : h1 ⊗ · · · ⊗ hn ⊗ f 7→ h1 S(f(1) ) ⊗ · · · ⊗ hn S(f(n) ) ⊗ f(n+1) .
We will use a “left” analogue of the Fourier transformation
F ′ : h ⊗ f1 ⊗ · · · ⊗ fn 7→ h(1) ⊗ h(2) f1 ⊗ · · · ⊗ h(n+1) fn ,
which is also invertible.

2.2

Dual algebras

Suppose H is a cocommutative Hopf algebra, and let X = H ∗ be its dual algebra (i.e., the
product on X is dual to the coproduct on H). Let us fix a linear basis {hi | i ∈ I} of H and
denote by {xi | i ∈ I} ⊂ X the set of dual functionals: hxi , hj i = δij , i, j ∈ I.
An arbitrary element x ∈ X can be presented as an infinite series in xi :

X
x=
hx, hi ixi .
i∈I

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P. Kolesnikov

The algebra X is a left and right module over H with respect to the actions given by
hxh, gi = hx, gS(h)i,

hhx, gi = hx, S(h)gi,

x ∈ X, h, g ∈ H.

(2.3)

The actions (2.3) turn X into a differential H-bimodule, i.e., (xy)h = (xh(1) )(yh(2) ), h(xy) =
(h(1) x)(h(2) y).

The operation ∆X : X → X ⊗ X := (H ⊗ H)∗ dual to the product on H is somewhat similar
to a coproduct. From the P
combinatorial point of view, X ⊗ X can be considered as the linear
αij xi ⊗ xj , αij ∈ k.
space of all infinite series
i,j∈I

In order to unify notations, we will use x(1) ⊗ x(2) for ∆X (x), x ∈ X. In particular, the
analogues of Fourier transforms
F −1 : x ⊗ y 7→ xS ∗ (y(1) ) ⊗ y(2) ,

F : x ⊗ y 7→ xy(1) ⊗ y(2) ,

x, y ∈ X,

act from X ⊗ X to X ⊗ X.
Definition 1. Let V be a linear space. A linear map π : X ⊗ X → V is said to be local, if
for almost all of the pairs (i, j) ∈ I 2 .

π(xi ⊗ xj ) = 0

¯
A local map π : X ⊗ X → V can be naturally continued to π
¯ : X ⊗ X → V . The map π
is continuous with respect to the topology on X ⊗ X defined by the following family of basic
neighborhoods of zero:
U ⊥ = {ξ ∈ X ⊗ X | hξ, U i = 0},

U ⊆ H ⊗ H,

dim U < ∞

(we assume V is endowed with discrete topology). Conversely, given a continuous linear map
X ⊗ X → V , its restriction to X ⊗ X is local.
For example, let us fix h1 , h2 ∈ H and consider the map x ⊗ y 7→ hx, h1 ihy, h2 i. It is clear
that this map is local. Obviously, every local map π : X ⊗ X → V is actually the “evaluation”
map
π(x ⊗ y) = eva (x, y) := (hx, ·i ⊗ hy, ·i ⊗ idV )(a)

(2.4)

for an appropriate a ∈ H ⊗ H ⊗ V .
Lemma 2. Suppose π : X ⊗ X → V is a local map, and denote πF = π
¯ ◦ F, πF −1 = π
¯ ◦ F −1 .
−1
−1
Then both πF and πF are local, πF = 0 implies π = 0, and πF = 0 implies π = 0.
Proof . Formally speaking, we can not use F −1 as an inverse of F since both F and F −1 are
not defined on the entire space X ⊗ X. But it is straightforward to check (see also [1]) that
X
X
∆X (x) = x(1) ⊗ x(2) =
xS(hi ) ⊗ xi =
xi ⊗ S(hi )x, x ∈ X,
i∈I

i∈I

so

πF(xi ⊗ xj ) = π
¯

X

xi (xj S(hk )) ⊗ xk

k∈I

=

X

!



=π
¯

X

k,l∈I

hxi , hl(1) ihxj , hl(2) hk iπ(xl ⊗ xk ).



hxi (xj S(hk )), hl ixl ⊗ xk 

k,l∈I

This is easy to deduce that if π = eva as in (2.4) then
πF(x ⊗ y) = π ′ (x ⊗ y),

π ′ = eva′ ,

a′ = (F ′ ⊗ idV )(a) ∈ H ⊗ H ⊗ V.

Since F ′ : H ⊗ H → H ⊗ H is invertible, a = 0 iff a′ = 0. Hence, π ′ = πF = 0 implies π = 0.
For πF −1 the proof is completely analogous.


Simple Finite Jordan Pseudoalgebras

2.3

5

Pseudoalgebras

In the exposition of the notion of pseudoalgebra we will preferably follow [1].
Hereinafter, H is a cocommutative Hopf algebra, e.g., H = U (g) or H = U (g)#k[Γ].
Definition 2 ([1]). Let P be a left H-module. A pseudoproduct is an H-bilinear map
∗ : P ⊗ P → (H ⊗ H) ⊗H P.
An H-module P endowed with a pseudoproduct ∗ is called a pseudoalgebra over H (or Hpseudoalgebra). If P is a finitely generated H-module, then P is said to be finite pseudoalgebra.
For every n, m ≥ 1, an H-bilinear map ∗ : P ⊗ P → (H ⊗ H) ⊗H P can be naturally expanded
to a map from (H ⊗n ⊗H P ) ⊗ (H ⊗m ⊗H P ) to (H ⊗n+m ⊗H P ):
(F ⊗H a) ∗ (G ⊗H b) = ((F ⊗ G) ⊗H 1)((∆[n] ⊗ ∆[m] ) ⊗H idP )(a ∗ b),

(2.5)

where F ∈ H ⊗n , G ∈ H ⊗m , a, b ∈ P .
This operation allows to consider long terms in P with respect to ∗.
One of the main features of a cocommutative bialgebra H is that symmetric groups Sn act
by H-module automorphisms on H ⊗n with respect to (2.1). The action of σ ∈ Sn is defined by
σ(h1 ⊗ · · · ⊗ hn ) = h1σ−1 ⊗ · · · ⊗ hnσ−1 .
Let us write down the obvious rules matching the action of Sn with the “expanded” pseudoproduct (2.5). For every A ∈ H ⊗n ⊗H P , B ∈ H ⊗m ⊗H P , τ ∈ Sn , σ ∈ Sm we have
((τ ⊗H idP )(A)) ∗ B = (¯
τ ⊗H idP )(A ∗ B),

(2.6)

where τ¯ ∈ Sn+m , k¯
τ = kτ for k = 1, . . . , n, k¯
τ = k for k = n + 1, . . . , n + m;
A ∗ ((σ ⊗H idP )(B)) = (σ+n ⊗H idP )(A ∗ B),

(2.7)

where iσ+n = i for i = 1, . . . , n, (n + j)σ+n = n + jσ for j = 1, . . . , m.
A pseudoproduct ∗ : P ⊗ P → (H ⊗ H) ⊗H P can be completely described by a family of
binary algebraic operations. Let P be an H-pseudoalgebra, X = H ∗ . Lemma 1 implies that for
every a, b ∈ P their pseudoproduct has a unique presentation of the form
X
a∗b=
(hi ⊗ 1) ⊗H ci ,
i

where {hi | i ∈ I} is a fixed basis of H. Consider the projections (called Fourier coefficients
of a ∗ b)
X
(a ◦x b) =
hx, S(hi )ici ∈ P,
i

for all x ∈ X. The x-products obtained have the following properties:
locality
(a ◦xi b) = 0

for almost all i ∈ I;

(2.8)

sesqui-linearity
(ha ◦x b) = (a ◦xh b),

(a ◦x hb) = h(2) (aS(h(1) )x b).

(2.9)

Note that the locality property does not depend on the choice of a basis in H: (2.8) means
that codim{x ∈ X | (a ◦x b) = 0} < ∞.

6

P. Kolesnikov

Remark 1 ([1]). In the case H = k[D], X ≃ k[[t]], where htn , Dm i = n!δn,m , the correspondence
between conformal n-products (n ≥ 0) and the pseudoproduct is provided by
a∗b=

X 1
((−D)n ⊗ 1) ⊗H (a ◦n b),
n!

n≥0

i.e., a ◦n b = a ◦tn b, n ≥ 0.
In the same way, one may define Fourier coefficients of an arbitrary
element A ∈ H ⊗n ⊗H P ,
P
(hi1 ⊗ · · · ⊗ hin−1 ⊗ 1) ⊗H a¯ı ,
n ≥ 2. By Lemma 1 A can be uniquely presented as A =
¯ı

¯ı = (i1 , . . . , in−1 ) ∈ I n−1 . By abuse of terminology, we will call these a¯ı ∈ P Fourier coefficients
of A.
There is a canonical way to associate an ordinary algebra A(P ) with a given pseudoalgebra P [1]. As a linear space, A(P ) coincides with X ⊗H P , and the product is given by
(x ⊗H a)(y ⊗H b) = S ∗ (x(1) )y ⊗H (a ◦x(2) b),

x, y ∈ X,

a, b ∈ P.

The algebra A(P ) obtained is called the annihilation algebra of P . If P is a torsion-free Hmodule then the structure of P can be reconstructed from A(P ) [1].
In the case of conformal algebras (H = k[D]), there is a slightly different construction
called coefficient algebra [7, 8, 15]. Suppose C is a conformal algebra and consider the space
Coeff C = k[t, t−1 ] ⊗k[D] C, where and D acts on k[t, t−1 ] as tn D = −ntn−1 . Denote tn ⊗k[D] a
by a(n), a ∈ C, n ∈ Z. The product on Coeff C is provided by
X n
(a ◦s b)(n + m − s),
n, m ∈ Z,
a, b ∈ C.
a(n)b(m) =
s
s≥0

An arbitrary conformal algebra can be embedded into a conformal algebra of formal power series
over its coefficient algebra [7].

2.4

Varieties of pseudoalgebras

Suppose Ω is a variety of ordinary algebras over a field of zero characteristic. Then Ω is defined
by a family of homogeneous polylinear identities. Such an identity can be written as
X
tσ (x1σ , . . . , xnσ ) = 0,
(2.10)
σ∈Sn

where each tσ (y1 , . . . , yn ) is a linear combination of non-associative words obtained from y1 . . . yn
by some bracketing.
Definition 3 ([13]). Let Ω be a variety of ordinary algebras defined by a family of homogeneous
polylinear identities of the form (2.10). Then set the Ω variety of pseudoalgebras as the class of
pseudoalgebras satisfying the respective “pseudo”-identities of the form
X
(σ ⊗H idC )t∗σ (x1σ , . . . , xnσ ) = 0,
(2.11)
σ∈Sn

where t∗σ means the same term tσ with respect to the pseudoproduct operation ∗.
If P is an Ω pseudoalgebra (or, in particular, conformal algebra) then its annihilation
(coefficient) algebra belongs to the Ω variety of ordinary algebras [13]. The converse is also
true for conformal algebras.

Simple Finite Jordan Pseudoalgebras

7

However, the class of Ω pseudoalgebras is not a variety in the ordinary sense: This class is
not closed under Cartesian products.
The main object of our study is the class of Jordan pseudoalgebras. Recall that the variety
of Jordan algebras is defined by the following identities:
ab = ba,

((aa)b)a = (aa)(ba).

(2.12)

In the polylinear form (if char k 6= 2, 3) the second identity of (2.12) can be rewritten as follows
(see, e.g., [20]):
[abcd] + [dbca] + [cbad] = {abcd} + {acbd} + {adcb}.
Here [. . . ] and {. . . } stand for the following bracketing schemes: [a1 . . . an ] = (a1 [a2 . . . an ]),
{a1 a2 a3 a4 } = ((a1 a2 )(a3 a4 )).
Therefore, an H-module P (over a cocommutative Hopf algebra H) endowed with a pseudoproduct ◦ is a Jordan pseudoalgebra iff it satisfies the following identities of the form (2.11):
a ◦ b = (σ12 ⊗H idP )(b ◦ a),
[a ◦ b ◦ c ◦ d] + (σ14 ⊗H idP )[d ◦ b ◦ c ◦ a] + (σ13 ⊗H idP )[c ◦ b ◦ a ◦ d]
= {a ◦ b ◦ c ◦ d} + (σ23 ⊗H idP ){a ◦ c ◦ b ◦ d} + (σ24 ⊗H idP ){a ◦ d ◦ c ◦ b},

(2.13)

where σij = (i j) are the transpositions from S4 .
As in the case of ordinary algebras, the natural relations hold between associative, Lie, and
Jordan pseudoalgebras. An associative pseudoalgebra P with respect to the new pseudoproduct
[a ∗ b] = a ∗ b − (σ12 ⊗H idP )(b ∗ a)
is a Lie pseudoalgebra denoted by P (−) [1]. Similarly, another pseudoproduct ◦ given by
a ◦ b = a ∗ b + (σ12 ⊗H idP )(b ∗ a)
makes P into a Jordan pseudoalgebra P (+) [13].
Example 1. Let H ′ be a Hopf subalgebra of H, and let P ′ be an H ′ -pseudoalgebra with respect
to a pseudoproduct ∗′ . Define a pseudoproduct on P = H ⊗H ′ P ′ by linearity:
X
g, h ∈ H,
(hhi ⊗ ggi ) ⊗H (1 ⊗H ′ ci ),
(h ⊗H ′ a) ∗ (g ⊗H ′ b) =
i

P
where a ∗′ b = (hi ⊗ gi ) ⊗H ′ ci , a, b, ci ∈ P ′ . The pseudoalgebra P obtained is called the current
i

′
pseudoalgebra CurH
H′ P .
In particular, k ⊂ H is a Hopf subalgebra of H. Hence, an ordinary algebra A gives rise to
current pseudoalgebra Cur A = CurH
k A.
′
It is clear that if P ′ is an Ω pseudoalgebra over H ′ then so is CurH
H′ P .

Example 2. Consider H = U (h), where h is a Lie algebra. Then the free left H-module H ⊗ H
equipped by pseudoproduct
(h ⊗ a) ∗ (g ⊗ b) = (hb(1) ⊗ g) ⊗H (1 ⊗ ab(2) ),

a, b, g, h ∈ H,

is an associative pseudoalgebra. The submodule W (h) = H ⊗ h is a subalgebra of the corresponding Lie pseudoalgebra (H ⊗ H)(−) .
Note that if h′ is a Lie subalgebra of h, then H ′ = U (h′ ) is a Hopf subalgebra of H, and
′
CurH
H ′ W (h ) is actually a subalgebra of W (h).

8

P. Kolesnikov

In particular, if h is the 1-dimensional Lie algebra then W (h) is just the Virasoro conformal
algebra [7].
Later we will use the classification of simple finite Lie pseudoalgebras [1]. Although the
results obtained in [1] are much more explicit, the following statements are sufficient for our
purposes.
Theorem 1 ([1]). A simple finite Lie pseudoalgebra L over H = U (h), dim h < ∞, k = C, is
isomorphic either to Cur g, where g is a simple finite-dimensional Lie algebra, or to a subalgebra
of W (h).

Theorem 2 ([1]). A simple Lie pseudoalgebra L over H = U (h)#k[Γ] which is finite over U (h)
(dim h < ∞, k = C) is a finite direct sum of isomorphic simple U (h)-pseudoalgebras such that
Γ acts on them transitively.


2.5

Conformal linear maps

Let H be a cocommutative Hopf algebra, and let M1 , M2 be two left H-modules. A map
ϕ : M1 → (H ⊗ H) ⊗H M2 is said to be (left) conformal linear if
ϕ(ha) = (1 ⊗ h)ϕ(a),

h ∈ H,

a ∈ M1 .

The set of all left conformal linear maps is denoted by Choml (M1 , M2 ). For M1 = M2 = M we
denote Choml (M, M ) = Cendl (M ).
For every H-modules M1 , M2 , the set Choml (M1 , M2 ) can be considered as an H-module
with respect to the action
hϕ(a) = (h ⊗ 1)ϕ(a),

h ∈ H,

ϕ ∈ Choml (M1 , M2 ),

a ∈ M1 .

For example, if P is a pseudoalgebra, a ∈ P , then the operator of left multiplication La : b 7→
a ∗ b, b ∈ P , belongs to Cendl (P ).
In order to unify notations, we will use ϕ ∗ a for ϕ(a), a ∈ M , ϕ ∈ Cendl (M ). One may
consider ∗ here as an H-bilinear map from Cendl (M ) ⊗ M to (H ⊗ H) ⊗H M . The relation (2.5)
allows to expand this map to
(H ⊗n ⊗H Cendl (M )) ⊗ (H ⊗m ⊗H M ) → H ⊗n+m ⊗H M.
The correspondence between ϕ ∗ a and (ϕ ◦x a) (x ∈ X) is given by
X
ϕ∗a=
(S(hi ) ⊗ 1) ⊗H (ϕ ◦xi a).
i∈I

The space Cendl (M ) can be also endowed with a family of x-products given by
(ϕ ◦x ψ) ◦y a = (ϕ ◦x(2) (ψ ◦S ∗ (x(1) )y a)),

ϕ, ψ ∈ Cendl (M ),

x, y ∈ X,

for a ∈ M .
The x-products (· ◦x ·) on Cendl (M ) satisfy (2.9), but (2.8) does not hold, in general.
To ensure the locality, it is sufficient to assume that M is a finitely generated H-module [1,
Section 10]. Therefore, Cendl (M ) for a finitely generated H-module M is an associative Hpseudoalgebra.
For a finite pseudoalgebra P , it is easy to rewrite the identity (2.13) using the operators of
left multiplication. Namely, this identity is equivalent to
La ∗ Lb ∗ Lc + (σ13 ⊗H id)(Lc ∗ Lb ∗ La ) + (σ123 ⊗H id)Lb∗(c∗a)
= La∗b ∗ Lc + (σ23 ⊗H id)(La∗c ∗ Lb ) + (σ13 ⊗H id)(Lc∗b ∗ La ),
where σ123 denotes the permutation (1 2 3) ∈ S4 .

Simple Finite Jordan Pseudoalgebras

3

9

Tits–Kantor–Koecher construction
for f inite Jordan pseudoalgebras

The general scheme described in [10, 11, 12, 18] provides an embedding of a Jordan algebra into
a Lie algebra. It is called the Tits–Kantor–Koecher (TKK) construction for Jordan algebras.
Let us recall the TKK construction for ordinary algebras. For a Jordan algebra j, the set
of derivations Der(j) is a Lie subalgebra of End(j) with respect to the commutator of linear
maps. Consider the (formal) direct sum S(j) = Der(j) ⊕ L(j), where L(j) is the linear space of
all left multiplications La : b 7→ ab, a ∈ j. It is well-known that [L(j), L(j)] ⊆ Der(j). Then the
space S(j) with respect to the new operation [·, ·] given by
[(La + D), (Lb + T )] = LDb − LT a + [La , Lb ] + [D, T ].
is a Lie algebra called the structure Lie algebra of j. Finally, consider
T(j) = j− ⊕ S0 (j) ⊕ j+ ,
where j± ≃ j, S0 (j) is the subalgebra of S(j), generated by Ua,b = Lab + [La , Lb ] ∈ S(j), a, b ∈ j.
Let us endow T(j) with the following operation:
[Σ, a− ] = (Σa)− ,

[a− , b+ ] = Ua,b ,

[a+ , b+ ] = [a− , b− ] = 0,
[Σ, a+ ] = (Σ∗ a)+ ,

[a− , Σ] = −(Σa)− ,

[a+ , Σ] = −(Σ∗ a)+ ,

∗
[a+ , b− ] = Ua,b
,

where Σ∗ = −La + D for Σ = La + D ∈ S(j). This operation makes T(j) to be a Lie algebra
called the TKK construction for j.
In the case of conformal algebras, a similar construction was introduced in [19] by making use
of coefficient algebras. We are going to get an analogue of TKK construction for finite Jordan
pseudoalgebras using the language of pseudoalgebras rather than annihilation algebras.
Definition 4. Let P be an H-pseudoalgebra. A conformal endomorphism T ∈ Cendl (P ) is said
to be a (left) pseudoderivation, if
T ∗ (a ∗ b) = (T ∗ a) ∗ b + (σ12 ⊗H idP )(a ∗ (T ∗ b))

(3.1)

for all a, b ∈ P . The set of all pseudoderivations of P we denote by Derl (P ).
In particular, if P is a finite pseudoalgebra then (3.1) is equivalent to [T ∗ La ] = LT ∗a , a ∈ P .
Lemma 3. Suppose that for some A ∈ (H ⊗ H) ⊗H Cendl (P ) the equality
A ∗ (a ∗ b) = (A ∗ a) ∗ b + (σ132 ⊗H id)(a ∗ (A ∗ b))
holds for all a, b ∈ P . Then all Fourier coefficients of A belong to in Derl (P ).
Proof . For every B ∈ H ⊗n+1 ⊗H M (M is an H-module), there exists a unique presentation
X
B=
(G¯ı ⊗ 1) ⊗H b¯ı ,
¯ı

where G¯ı form a linear basis of H ⊗n (see Lemma 1). By Bx1 ,...,xn , xi ∈ X, we denote the
expression
X
hx1 ⊗ · · · ⊗ xn , G¯ı ib¯ı .
¯ı

10

P. Kolesnikov

It is clear that the map (x1 , . . . , xn ) 7→ Bx1 ,...,xn is polylinear. If we fix an arbitrary set of n − 2
arguments, then the map X ⊗ X → M obtained is local in the sense of Definition 1.
P
P
P
(hk ⊗ 1) ⊗H dijk . Then
(hj ⊗ 1) ⊗H cj , Di ∗ cj =
Let A = (hi ⊗ 1) ⊗H Di , a ∗ b =
j∈I

i∈I

Di ∗ (a ∗ b) =

X

k∈I

(hk ⊗ hj ⊗ 1) ⊗H dijk ,

(3.2)

(hi hk(1) ⊗ hk(2) ⊗ hj ⊗ 1) ⊗H dijk .

(3.3)

j,k∈I

A ∗ (a ∗ b) =

X

i,j,k∈I

Compare (3.2) and (3.3) to get
(A ∗ (a ∗ b))x,y,z =

X

hx(1) , hi i(Di ∗ (a ∗ b))x(2) y,z .

(3.4)

X

hx(1) , hi i((Di ∗ a) ∗ b)x(2) y,z ,

(3.5)

i∈I

In the same way,
((A ∗ a) ∗ b)x,y,z =

i∈I

X




hx(1) , hi i (σ12 ⊗H id)(a ∗ (Di ∗ b)) x
(σ132 ⊗H id)(a ∗ (A ∗ b)) x,y,z =

(2) y,z

.

(3.6)

i∈I

The relations (3.4)–(3.6) together with Lemma 2 imply
π(x, y, z) =

X
i∈I



hx, hi i Di ∗ (a ∗ b) − (Di ∗ a) ∗ b) − (σ12 ⊗H id)((Di ∗ a) ∗ b) y,z = 0.

It means that
Di ∗ (a ∗ b) − (Di ∗ a) ∗ b) − (σ12 ⊗H id)(a ∗ (Di ∗ b)) = 0.



Lemma 4. For a finite pseudoalgebra P the set of all pseudoderivations is a subalgebra of the
Lie pseudoalgebra Cendl (P )(−) .
Proof . Let D1 , D2 ∈ Derl (P ), i.e., [Di ∗ La ] = LDi ∗a for a ∈ P , i = 1, 2.
Since Cendl (P )(−) satisfies Jacobi identity,
[[D1 ∗ D2 ] ∗ La ] = [D1 ∗ [D2 ∗ La ]] − (σ12 ⊗H id)([D2 ∗ [D1 ∗ La ]])
= [D1 ∗ LD2 ∗a ] − (σ12 ⊗H id)([D2 ∗ LD1 ∗a ])
= LD1 ∗(D2 ∗a) − (σ12 ⊗H id)LD2 ∗(D1 ∗a) = L[D1 ∗D2 ]∗a .
Hence, for every a, b ∈ P we have
[D1 ∗ D2 ] ∗ (a ∗ b) = ([D1 ∗ D2 ] ∗ a) ∗ b + (σ132 ⊗H id)(a ∗ ([D1 ∗ D2 ] ∗ b)).
Lemma 3 implies that [D1 ◦x D2 ] ∈ Derl (P ) for all x ∈ X.



Lemma 5. Let J be a finite Jordan pseudoalgebra, and let L(J) be the H-submodule of Cendl (J)
generated by {La | a ∈ J}. Then [La ◦x Lb ] is a pseudoderivation for every x ∈ X, i.e.,
L′ (J) ⊆ Derl (J).

Simple Finite Jordan Pseudoalgebras

11

Proof . The following relation is easy to deduce from (2.13):
La ∗ Lc∗d + (σ13 ⊗H id)(Ld ∗ Lc∗a ) + (σ12 ⊗H id)(Lc ∗ La∗d )
= (σ123 ⊗H id)(Lc∗d ∗ La ) + La∗c ∗ Ld + (σ23 ⊗H id)(La∗d ∗ Lc ).
So by (2.6), (2.7)
[La ∗ Lc∗d ] = [La∗c ∗ Ld ] − (σ12 ⊗H id)[Lc ∗ La∗d ].

(3.7)

It is sufficient to prove that for every a, b, c ∈ J we have [[La ∗ Lb ] ∗ Lc ] = L[La ∗Lb ]∗c , i.e.,
La ∗ Lb ∗ Lc − (σ132 ⊗H id)(Lc ∗ La ∗ Lb ) + (σ13 ⊗H id)Lc ∗ Lb ∗ La
− (σ12 ⊗H id)Lb ∗ La ∗ Lc = La∗(b∗c) − (σ12 ⊗H id)Lb∗(a∗c) .

(3.8)

Indeed,
La ∗ Lb ∗ Lc + (σ13 ⊗H id)(Lc ∗ Lb ∗ La ) = −(σ123 ⊗H id)Lb∗(c∗a)
+ La∗b ∗ Lc + (σ23 ⊗H id)(La∗c ∗ Lb ) + (σ13 ⊗H id)(Lc∗b ∗ La ),

(3.9)

(σ12 ⊗H id)(Lb ∗ La ∗ Lc ) + (σ132 ⊗H id)(Lc ∗ La ∗ Lb )
= (σ12 ⊗H id)(Lb ∗ La ∗ Lc ) + (σ12 σ13 ⊗H id)(Lc ∗ La ∗ Lb )
= −(σ12 σ123 ⊗H id)La∗(c∗b) + (σ12 ⊗H id)(Lb∗a ∗ Lc )
+ (σ12 σ23 ⊗H id)(Lb∗c ∗ La ) + (σ12 σ13 ⊗H id)(Lc∗a ∗ Lb )
= −(σ23 ⊗H id)La∗(c∗b) + (σ12 ⊗H id)(Lb∗a ∗ Lc )
+ (σ123 ⊗H id)(Lb∗c ∗ La ) + (σ132 ⊗H id)(Lc∗a ∗ Lb ).

(3.10)

Subtracting (3.9) from (3.10) and using commutativity La∗b = (σ12 ⊗H id)Lb∗a , we ob
tain (3.8).
Definition 5. Let J be a finite Jordan pseudoalgebra. The formal direct sum of H-modules
S(J) = L(J) ⊕ Derl (J)
endowed with the pseudoproduct
[(La + D) ∗ (Lb + T )] = LD∗b − (σ12 ⊗H id)LT ∗a + [La ∗ Lb ] + [D ∗ T ]

(3.11)

is called the structure Lie pseudoalgebra of J.
It is straightforward to check that the (pseudo) anticommutativity and Jacobi identities hold
for (3.11).
Consider the elements Ua,b = La∗b + [La ∗ Lb ] ∈ (H ⊗ H) ⊗H S(J), a, b ∈ J. By U(a◦x b) =
L(a◦x b) + [La ◦x Lb ], x ∈ X, we denote the Fourier coefficients of Ua,b . The linear space S0 (J)
generated by the set {U(a◦x b) | a, b ∈ J, x ∈ X} is an H-submodule of S(J).
Proposition 1. The H-module S0 (J) is closed under the pseudoproduct (3.11), i.e., S0 (J) is
a Lie pseudoalgebra.
Proof . Let us calculate [Ua,b ∗ Uc,d ], a, b, c, d ∈ J. Denote D = [La ∗ Lb ], A = a ∗ b. Then
[Ua,b ∗ Lc∗d ] = [LA ∗ Lc∗d ] + L(D∗c)∗d + (σ132 ⊗H id)Lc∗(D∗d) , [Ua,b ∗ [Lc ∗ Ld ]] = [LA ∗ [Lc ∗ Ld ]] +
[LD∗c ∗ Ld ] + (σ132 ⊗H id)[Lc ∗ LD∗d ]. Therefore,
[Ua,b ∗ Uc,d ] = [LA ∗ Lc∗d ] + [LA ∗ [Lc ∗ Ld ]] + UD∗c,d + (σ132 ⊗H id)Uc,D∗d .
From the first summand of the right-hand side we obtain [LA ∗ Lc∗d ] = [LA∗c ∗ Ld ] − (σ132 ⊗H
id)[Lc ∗ LA∗d ] by using (3.7). Moreover, [LA ∗ [Lc ∗ Ld ]] = L(A∗c)∗d − (σ132 ⊗H id)Lc∗(A∗d) . Hence,
[Ua,b ∗ Uc,d ] = UA∗c,d + UD∗c,d + (σ132 ⊗H id)(Uc,D∗d − Uc,A∗d ).



12

P. Kolesnikov

∗ = −L
∗
Denote Ua,b
a∗b + [La ∗ Lb ], a, b ∈ J. Note that Ua,b = −(σ12 ⊗H id)Ub,a , so all Fourier
∗
coefficients of Ua,b lie in S0 (J). If J is a Jordan pseudoalgebra and J 2 = J, i.e., every a ∈ J lies
in the subspace generated by the set {(b ◦x c) | b, c ∈ J, x ∈ X}, then S0 (J) ⊃ L(J).
Indeed, for every a, b ∈ J we have Ua,b + (σ12 ⊗H idJ )Ub,a = 2La∗b , so L(a◦x b) ∈ S0 (J). Hence,
L(J) = L(J 2 ) ⊂ S0 (J).
Let us consider the direct sum of H-modules

T(J) = J − ⊕ S0 (J) ⊕ J + ,
where J + and J − are isomorphic copies of J. Given a ∈ J (or A ∈ H ⊗n ⊗H J), we will denote
by a± (or A± ) the image of this element in J ± (or H ⊗n ⊗H J ± ). Define a pseudoproduct on
T(J) by the following rule: for a± , b± ∈ J ± , Σ ∈ S0 (J) set
∗
[a+ ∗ b− ] = Ua,b
,

[a− ∗ b+ ] = Ua,b ,

[a− ∗ Σ] = −(σ12 ⊗H id)(Σ ∗ a)− ,
+

∗

+

[a ∗ Σ] = −(σ12 ⊗H id)(Σ ∗ a) ,

[a+ ∗ b+ ] = [a− ∗ b− ] = 0,
[Σ ∗ a− ] = (Σ ∗ a)− ,
+

∗

(3.12)

+

[Σ ∗ a ] = (Σ ∗ a) .

Set the pseudoproduct on S0 (J) to be the same as (3.11). Here we have used Σ∗ = −La + D for
Σ = La + D ∈ S(J).
Denote the projections of T(J) on J + , J − , S0 (J) by π+ , π− , π0 , respectively. It is straightforward to check that T(J) is a Lie pseudoalgebra. This is an analogue of the Tits–Kantor–Koecher
construction for an ordinary Jordan algebra.
Note that the structure pseudoalgebra is a formal direct sum of the corresponding H-modules,
so the condition
Σ∗b=0

for all b ∈ J

does not imply Σ = 0 in S(J). However, if Σ = La +D ∈ S(J) and [Σ∗b− ] = [Σ∗b+ ] = 0 in T(J)
for all b ∈ J, then a ∗ b + D ∗ b = 0 and −a ∗ b + D ∗ b = 0 by (3.12). Therefore, a ∗ b = D ∗ b = 0
for all b ∈ J, i.e., Σ = 0 in S(J).
Proposition 2. Let J be a simple finite Jordan pseudoalgebra. Then L = T(J) is a simple
finite Lie pseudoalgebra.
Proof . Suppose that there exists a non-zero proper ideal I ✁ L. Let
J± = {a ∈ J | a± = π± (b) for some b ∈ I}.
Since J 2 = J, we have L ⊃ L(J). Hence, J± ✁ J.
0 = {a ∈ J | a± ∈ I ∩ J ± } are also some ideals in J.
Analogously, J±
0 = J 0 = 0). Since I 6= 0, there exists
1) Consider the case J+ = J− = 0 (hence, J+
−
0 = 0, so Σ = 0 as we have shown
Σ = Lb + D ∈ S0 (J) ∩ I, Σ 6= 0. But [Σ ∗ J ± ] ⊆ H ⊗2 ⊗H J±
above, which is a contradiction.
0 = 0. Then for each a ∈ J there exists a+ + Σ + d− ∈ I. Consider
2) Let J+ = J, J−
∗
∗
0
= 0.
[[(a+ + Σ + d− ) ∗ b− ] ∗ c− ] = [(Ua,b
+ (Σ ∗ b)− ) ∗ c− ] = (Ua,b
∗ c)− ∈ H ⊗3 ⊗H J−

For every a, b, c ∈ J we have
−La∗b ∗ c + [La ∗ Lb ] ∗ c = 0.
(3.13)
P
P
If a ∗ b = (hi ⊗ 1) ⊗H (a ◦xi b), then b ∗ a = (1 ⊗ hi ) ⊗H (a ◦xi b) by commutativity.
i
i
P
Therefore, La∗b = (hi ⊗ 1) ⊗H La◦xi b = (σ12 ⊗H idL(J) )Lb∗a . By the definition of commutator,
i

Simple Finite Jordan Pseudoalgebras

13

(σ12 ⊗H idJ )([La ∗ Lb ] ∗ c) = −[Lb ∗ La ] ∗ c. Relation (3.13) implies −Lb∗a ∗ c + [Lb ∗ La ] ∗ c = 0
by symmetry. Hence, 0 = (σ12 ⊗H idJ )(−Lb∗a ∗ c + [Lb ∗ La ] ∗ c) = −La∗b ∗ c − [La ∗ Lb ] ∗ c.
Compare the last relation with (3.13) to get La∗b ∗ c = 0 for all a, b, c ∈ J. Then the condition
J 2 = J implies L(J) = 0, which is a contradiction.
0 = 0 is completely analogous.
3) The case J− = J, J+
0 is zero, then at least one of the ideals J ✁ J has to be zero,
Hereby, if either of the ideals J±
±
0
0 = J, i.e., I ⊃ J + , J − . Since the whole pseudoalgebra L is
which is impossible. Hence, J+ = J−
generated by J + ∪ J − , we have I = L.


4

Structure of simple Jordan pseudoalgebras

We have shown (Proposition 2), that for a simple finite Jordan pseudoalgebra J its TKK construction L = T(J) is a simple finite Lie pseudoalgebra. This allows to describe simple Jordan
pseudoalgebras using the classification of simple Lie pseudoalgebras [1].

4.1

The case H = U (h)

Throughout this subsection, H is the universal enveloping Hopf algebra of a finite-dimensional
Lie algebra h over C.
Proposition 3. Let J be a simple finite Jordan H-pseudoalgebra. Then the TKK construction T(J) is isomorphic to the current algebra Cur g over a simple finite-dimensional Lie algebra g.
Proof . If J is a simple finite Jordan H-pseudoalgebra, then T(J) is a simple finite Lie pseudoalgebra. Hence, either T(J) = Cur g, where g is a simple finite-dimensional Lie algebra, or T(J)
is a subalgebra in W (h) (see Theorem 1 and Example 2). The second case could not be realized
since by [1, Proposition 13.6] the pseudoalgebra W (h) does not contain abelian subalgebras.
This is not the case for T(J).

It remains to show that if T(J) = J + ⊕S0 (J)⊕J − = Cur g then J is the current pseudoalgebra
over a simple finite-dimensional Jordan algebra.
Suppose e1 , . . . , en is a basis of h. Then the set of monomials
(α1 )

e(α) = e1

. . . en(αn ) ,

α = (α1 , . . . , αn ) ∈ Zn ,

αi ≥ 0,

(α )

where ei i = α1i ! eαi , is a basis of H. In order to simplify notation, we assume e(α) = 0 whenever
α contains a negative component.
Denote |α| = α1 + · · · + αn . We will use the standard deg-lex order on the set of monomials
of the form e(α) : e(α) ≤ e(β) if and only if α ≤ β, i.e., either |α| < |β| or |α| = |β| and α is
lexicographically less than β.
P
Suppose the multiplication rule in H is given by e(α) e(β) = γµα,β e(µ) . It is also useful to set
µ

γµα,β = 0 if either of α, β, µ contains a negative component.
P
The standard coproduct on H is easy to compute in this notation: ∆(e(α) ) = e(α−ν) ⊗ e(ν) .
ν

Theorem 3. Let J be a simple finite Jordan pseudoalgebra over H = U (h), where h is a finitedimensional Lie algebra over the field C. Then J is isomorphic to the current algebra Cur j over
a finite-dimensional simple Jordan algebra j.

Lemma 6. Let C
P
P= Cur g = H ⊗ g. Consider an arbitrary pair of elements a, b ∈ C, a =
e(α) ⊗ aα , b = e(β) ⊗ bβ , aα , bβ ∈ g. If [a ∗ b] = 0, then [aα bβ ] = 0 in g for all α, β.
α

β

14

P. Kolesnikov

Proof . Straightforward computations show that
X




[a ∗ b] =
(−1)|β−ν| γµα,β−ν e(µ) ⊗ 1 ⊗H e(ν) ⊗ [aα bβ ] .

(4.1)

If [a ∗ b] = 0 then (4.1) implies that for every ν, µ we have
X
(−1)|β−ν| γµα,β−ν [aα bβ ] = 0.

(4.2)

α,β,ν,µ

α,β

Put ν = β max in (4.2) (i.e., bν 6= 0, but bβ = 0 for all β > ν). We obtain

P

γνα,0 [aα bν ] = 0

α

for each µ. However,
(
0, µ 6= α,
γµα,0 =
1, µ = α,

hence, [aα bβ max ] = 0 for each α.
To finish the proof, use the induction on β. Suppose that [aµ bβ ] = 0 for all µ and β > β0 .
P α,0
Let us show that [aµ bβ0 ] = 0. Put ν = β0 . Relation (4.2) implies 0 =
γµ [aα bβ0 ] +
α
P
(−1)|ν−β| γµα,β−ν [aα bβ ]. The second summand is equal to zero by the inductive assumption.

β>β0

So we have [aα bβ0 ] = 0 for each α.



Now, let L = J + ⊕ S0 (J) ⊕ J − = P
Cur g. By j±
0 we denote the spaces spanned by all
e(α) ⊗ aα ∈ J ± . Lemma 6 implies the spaces j±
coefficients aα ∈ g ingoing in the sums
0 are
α

±
±
±
Abelian subalgebras of g such that [H ⊗ j±
0 ∗ J ] = 0. Moreover, H ⊗ j0 ⊇ J .

Lemma 7. Let L = T(J) = Cur g, where g is a finite-dimensional Lie algebra. Suppose that
there are no non-zero ideals I ✁ L such that π ± (I) = 0. Then J ± = H ⊗ j±
0 , respectively.
Proof . It is enough to consider the “+” case. Consider an arbitrary element a ∈ H ⊗ j+
0,
+
−
+
0
a = π+ (a) + π0 (a) + π− (a). Denote J0 = π− (H ⊗ j0 ), J0 = π0 (H ⊗ j0 ).
For every b ∈ J + we have 0 = [a ∗ b] = [π+ (a) ∗ b] + [π0 (a) ∗ b] + [π− (a) ∗ b]. Since [π0 (a) ∗ b] ∈
H ⊗2 ⊗H S0 (J), [π− (a) ∗ b] ∈ H ⊗2 ⊗H J + , [π+ (a) ∗ b] = 0, then
[J0− ∗ J + ] = [J00 ∗ J + ] = 0.

(4.3)

Given H-submodules A, B ⊆ L, denote by [A · B] ⊆ L the H-module spanned (over C) by all
Fourier
from [A ∗ B]. By [Aω · B] we denote the sum of H-modules
P n coefficients of0 all elements n+1
[A · B], where [A · B] = B, [A
· B] = [A · [An · B]].

n≥0

For example, [S0 (J)ω ·J0− ] ⊆ J − . Moreover, the Jacobi identity and (4.3) imply [J + ∗[S0 (J)ω ·
is also easy to note that [S0 (J) ∗ [S0 (J)ω · J0− ]] ⊆ H ⊗2 ⊗H [S0 (J)ω · J0− ]. Since
· J0− ]] = 0, then I = [S0 (J)ω · J0− ] is a proper ideal of L, I ⊇ J0− and π + (I) = 0.
Hence, I = 0, and J0− = 0.
Further, let us consider

J0− ]] = 0. It
[J − ∗ [S0 (J)ω

I = [S0 (J)ω · J00 ] + [S0 (J)ω · [J − · J00 ]] ⊆ S0 (J) ⊕ J − .

(4.4)

It follows from (4.3) that [J + ∗ [S0 (J)ω · J00 ]] = 0. Moreover, [J + ∗ [S0 (J)ω · [J − · J00 ]]] ⊆
H ⊗2 ⊗H [S0 (J)ω · J00 ]]. Therefore, [J + · I] ⊆ I. Since [S0 (J) · I] ⊆ I by construction, and
[J − · I] ⊆ I by the Jacobi identity, the ideal (4.4) is proper in L, so J00 = 0.
+
+
+
We have proved that π− (H ⊗ j+

0 ) = π0 (H ⊗ j0 ) = 0. Thus, J = H ⊗ j0 .

Simple Finite Jordan Pseudoalgebras

15

Hence, under the conditions of Lemma 7 one has J = H ⊗ j, j ≃ j±
0 . Now it is necessary
to show that the Jordan pseudoproduct on J may be restricted to an ordinary Jordan product
on j.
Proposition 4. Let J be a finite Jordan H-pseudoalgebra such that Annl (J) := {a ∈ J | a ∗ J =
0} = 0. Assume that J = H ⊗ j, where j is a linear space. If L = T(J) = Cur g then j has
a structure of ordinary Jordan algebra such that g ≃ T(j).
Proof . Let a, b ∈ J be some elements of the form a = 1 ⊗ α, b = 1 ⊗ β, α, β ∈ j. Then
2La∗b = [a− ∗ b+ ] + (σ12 ⊗H id)[b− ∗ a+ ] = (1 ⊗ 1) ⊗H (1 ⊗ [α− β + ] + 1 ⊗ [β − α+ ]) (here α± denote
the images of α ∈ j in j±
0 ).
Thus, La∗b = (1 ⊗ 1) ⊗H (1 ⊗ s(α, β)) ∈ L, where s(α, β) ∈ [j− j+ ] ⊆ g. Therefore, L(a◦tν b) = 0
for ν = (ν1 , . . . , νn ) > (0, . . . , 0). Here we have used the notation tν = tν11 . . . tνnn for basic
functionals in X = H ∗ .
Since Lx = 0 implies x = 0, we have
a ∗ b = (1 ⊗ 1) ⊗H c,
c ∈ J.
(4.5)
P (µ)
Suppose that c =
e ⊗ γµ , γµ ∈ j. Assume that the maximal µ = µmax such that γµ 6= 0 is
µ

a multi-index greater than (0, . . . , 0). Then
[(1 ⊗ s(α, β)) ◦tµmax (1 ⊗ δ − )] = 0

(4.6)

for all δ ∈ j. On the other hand, [(1 ⊗ s(α, β)) ◦tµmax (1 ⊗ δ − )] = (c ◦tµmax (1 ⊗ δ))− . It is easy to
see that the relations (4.5), (4.6) and the axioms of a pseudoalgebra imply (c ◦tµmax (1 ⊗ δ)) =
((1 ⊗ γµmax ) ◦ε (1 ⊗ δ)) = 0, i.e., L1⊗γµmax = 0. Thus, γµmax = 0, which is a contradiction.
We have proved that (1⊗α)∗(1⊗β) = (1⊗1)⊗H (1⊗γ(α, β)), γ(α, β) ∈ j. This relation leads
to an ordinary product on j defined by the rule α · β = γ(α, β). Then the pseudoalgebra J is
a current pseudoalgebra over j, and (j, ·) is necessarily a simple finite-dimensional Jordan algebra.
To complete the proof, it is enough to note that T(Cur j) ≃ Cur T(j). For finite-dimensional Lie
algebras g1 , g2 the condition Cur g1 ≃ Cur g2 implies g1 ≃ g2 .

Proof of Theorem 3. Let J be a simple finite Jordan pseudoalgebra. Proposition 3 implies
that L = T(J) ≃ Cur g, where g is a simple finite-dimensional Lie algebra. By Lemma 7,
J = H ⊗ j. Since L satisfies the conditions of Proposition 4, we have J ≃ Cur j, T(j) = g, where
j is a simple finite-dimensional Jordan algebra.

Corollary 1 ([19]). A simple finite Jordan conformal algebra is isomorphic to the current
conformal algebra over a simple finite-dimensional Jordan algebra.


4.2

The case H = U (h)#C[Γ]

If J is a pseudoalgebra over H = U (h)#C[Γ] then it is in particular a pseudoalgebra over U (h).
The structure of H-pseudoalgebra on J is completely encoded by U (h)-pseudoalgebra structure
and by the action of Γ on U (h), see [1, Section 5] for details.
Theorem 4. Let J be a simple Jordan pseudoalgebra over H = U (h)#C[Γ], dim h < ∞, which
is a finitely generated U (h)-module. Then
J≃

m
M

CurU (h) ji ,

i=1

where ji are isomorphic finite-dimensional simple Jordan algebras, and Γ acts transitively on the
family {CurU (h) ji : i = 1, . . . , m}.

16

P. Kolesnikov

Proof . By Proposition 2 L = T(J) is a simple H-pseudoalgebra, and it is clear that L is
m
L
CurU (h) gi
a finitely generated U (h)-module. Theorem 2 and Proposition 3 imply that L =
i=1

where CurU (h) gi = Curi are isomorphic simple current Lie U (h)-pseudoalgebras, and Γ acts on
them transitively.
m
L
˜, where g
˜=
gi . The H-pseudoalgebra L could be considered as an
Hence, L = CurU (h) g
i=1

U (h)-pseudoalgebra endowed with an action of Γ on it which is compatible with that of U (h):
g(ha) = hg (ga), h ∈ U (h), a ∈ L, g ∈ Γ.
˜. The condition of Lemma 7 holds for
Consider L as the current U (h)-pseudoalgebra over g
this L. Indeed, if I is an ideal of the U (h)-pseudoalgebra L and π ± (I) = 0, then ΓI is a proper
ideal of L (as of an H-pseudoalgebra) such that its projections onto J ± are zero. Moreover,
if J as an H-pseudoalgebra has no non-trivial (left) annihilator Annl (J) then so is J as an
U (h)-pseudoalgebra (see [1, Corollary 5.1]).
Therefore, the same arguments as in the proof of Proposition 4 show that J = CurU (h) ˜j,
where ˜j is a finite-dimensional Jordan algebra.
The explicit expression [1, equation (5.7)] for pseudoproduct over H shows that for every
x ∈ X = U (h)∗ , g ∈ Γ, a, b ∈ J we have
(a ◦x⊗g∗ b) = (a ◦(x⊗1)g b) = (ga ◦x b),
where hg ∗ , γi = δg,γ , γ ∈ Γ. Hence, the following relation between Fourier coefficients of Ua,b
holds: U(a◦x⊗g∗ b) = U(ga◦x b) . Here in the left- and right-hand sides we state Fourier coefficients
over H ∗ and X, respectively. Therefore, the relations between the H-module S0 (H J) and the
U (h)-module U (h) S0 (J) are the same as between H-module H J and U (h)-module U (h) J.
m
L
˜ = T(˜j). Hence, ˜j =
ji , gi = T(ji ), where ji are simple Jordan algebras.
Now it is clear that g
So, J =

m
L

i=1

U (h)

Cur

ji , and Γ necessarily acts on these current algebras transitively.



i=1

Acknowledgements
I am very grateful to L.A. Bokut, I.V. L’vov, E.I. Zel’manov, and V.N. Zhelyabin for their
interest in the present work and helpful discussions. It is my pleasure to appreciate the efforts
of the referees, whose suggestions and comments helped me to make the paper readable.
The work was partially supported by SSc-344.2008.1. I gratefully acknowledge the support
of the Pierre Deligne fund based on his 2004 Balzan prize in mathematics, and Novosibirsk City
Administration grant of 2008.

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